From Special Relativity to Feynman Diagrams.pdf

The previous equations using the definition of

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order and can thus be neglected. The previous equations, using the definition of Poisson brackets, become: δ q i = Q i q i = − δθ r G r p i = − δθ r { q i , G r } , (8.92) δ p i = δ p i = P i p i = δθ r G r q i = − δθ r { p i , G r }; (8.93) δ H = H H = − δθ r G r t . (8.94) Accordingly, the quantity δθ r G r is called infinitesimal generator of the canonical transformation and the G r ’s build a basis of generators. Let us now consider a dynamic variable function of P i , q i and let us compute its transformation under an infinitesimal canonical transformation: δ f = f ( P + δ P , q + δ q ) f ( P , q ) = f P i δ p i + f q i δ q i = − δθ r f q i G r p i + f p i G r q i = − δθ r { f , G r } , (8.95) where, by the same token as before, we have approximated G r P i by G r p i and f P i by f p i since they are multiplied by an infinitesimal quantity. It is easy to verify
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8.4 Canonical Transformations and Conserved Quantities 229 that under the infinitesimal canonical transformation, using ( 8.92 ), ( 8.93 ) and the Jacobi identity ( 8.74 ), we have δ { q i , p j } = 0 . (8.96) That means that the fundamental canonical brackets between the Lagrangian coor- dinates and conjugate momenta are left invariant under an infinitesimal canonical transformation and therefore also by finite ones. It is important to observe that the time evolution of the dynamic system, i.e. the correspondence between the canonical variables computed at a time t and those evaluated at a later time t > t, can be considered as a particular canonical trans- formation whose infinitesimal generator is the Hamiltonian . Let us indeed consider the change of the canonical coordinates when the time is increased from t to t + dt : q i ( t ) q i ( t ) = q i ( t + dt ) q i ( t ) + dt ˙ q i ( t ), p i ( t ) p i ( t ) = p i ( t + dt ) p i ( t ) + dt ˙ p i ( t ). It is easy to show that the infinitesimal generator of this transformation is H . In fact if we identify: ( q , P ) = q i P i + dt H ( p , q , t ), and use ( 8.92 ), ( 8.93 ), upon identifying G = − H and δθ = dt , we find: δ q i = − dt G p i = dt H p i = dt ˙ q i , δ p i = dt G q i = − dt H q i = dt ˙ p i , where we have used the Hamilton equations ( 8.66 ). We may therefore state that The Hamiltonian is the infinitesimal generator of the time translations. In other words H generates the time evolution of the dynamic system. We further note that if we compute the Poisson brackets of the canonical variables with the Hamiltonian we find: { q i , H } = H p i ; { p i , H } = − H q i , (8.97) so that the Hamilton equations of motion ( 8.66 ) can be also written as follows: ˙ q i = { q i , H }; ˙ p i = { p i , H } . (8.98) 8.4.1 Conservation Laws in the Hamiltonian Formalism In the Lagrangian formalism the conservation laws were derived by requiring the symmetry transformations on the Lagrangian coordinates to leave the functional
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230 8 Lagrangian and Hamiltonian Formalism form of the Lagrangian invariant, modulo an additional total derivative. Applying this requirement to translations in space and time, and to rotations, we derived the conservation laws for energy, linear momentum and angular momentum.
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